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	\begin{document}
	
	\begin{table}[htbp] 
  \centering
  \caption{\bf Sharpe Ratios for Local-PCA: }
  \centering
   {\bf  In-Sample vs Out-of-Sample ($T=24$, Tangency - pa)}
\medskip
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccccccc}
  \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    &   - &    0.579 &  0.601 &  0.127 & -0.181 & -0.359 & -0.328&  -0.294 & -0.240 & -0.236 & -0.279 \\ 
  local-PCA (bal) &   0.674  & 0.891 &   0.602 &  0.302  & 0.271 &  0.533 &  0.443 &  0.275 &  0.489 &  0.229  &  0.266 \\
local-PCA (unbal) &   0.633 &  0.646 & 0.509  &  0.434  &  0.596 &  0.427 &  0.407  &  0.499 &  0.407   & 0.673 & 0.674 \\
  \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    & -  &  
      0.459 & 0.520 &  0.522 &  0.625 &  0.661 &  0.667 &  0.674 & 0.674 &  0.682 &  0.685 \\
  local-PCA (bal) &   1.252  &  0.690 & 0.957 &  1.042 &  1.169 &  1.301 &  1.487 &  1.625 & 1.739 &  1.881 & 2.039 \\
  local-PCA (unbal) &    1.013  & 0.784 & 1.06 & 1.187 &  1.363  & 1.487 &  1.610 &  1.716 &  1.932  &  2.015 &  2.160  \\ \hline 
    \end{tabular}
        \parbox{6.75in}{\footnotesize{This  table presents the  time-average (across $480-2T$ windows) of the time series for the annualized Sharpe Ratios  
         both in-sample (bottom panel) and out-of-sample (top panel) for the Tangency Portfolio,
    correspondingly to  the (time-varying) estimated number of factors $\tilde{r}$   (column two), described in Section~\ref{emp_number_tv}, and to the values  one to $ 10 $ (columns 3-12), respectively.  The local-PCA is evaluated with $T=24$. The constant-PCA results do not depend on $T=24$, and are identical to the $T=120$ table. Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table4OAT24}
\end{table}


 \begin{table}[htbp] 
  \centering
  \caption{\bf Sharpe Ratios for Local-PCA: } 
  \centering
{\bf    In-Sample vs Out-of-Sample  ($T=120$, Tangency - pa)}
\medskip
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccccccc}
  \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    &   - &    0.579 &  0.601 &  0.127 & -0.181 & -0.359 & -0.328&  -0.294 & -0.240 & -0.236 & -0.279 \\ 
  local-PCA (bal) &
      0.208 &  0.408 & 0.604 &  0.489 &  0.386 &  0.313 &  0.252 &  0.373 &  0.267 &  0.194 &  0.208 \\
local-PCA (unbal) & 
   0.137 &   0.521 &  0.211 &  0.230 &   0.265 &   0.179 &   0.223 &   0.243 &  0.197 &   0.144 &  0.137 \\
    \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA    & -  & 
     0.459 & 0.520 &  0.522 &  0.625 &  0.661 &  0.667 &  0.674 & 0.674 &  0.682 &  0.685 \\
    local-PCA (bal) & 
      1.014 &   0.477 &  0.500 & 0.619 &  0.730 & 0.816 &  0.836 &  0.879 &  0.928 &  0.942 &  0.960  \\
   local-PCA (unbal) & 
       0.960 & 0.650 & 0.673 &  0.712 &  0.806 &  0.859 & 0.903 & 0.934 &  0.954 &  0.973 & 1.014 \\  \hline
    \end{tabular}
        \parbox{6.75in}{\footnotesize{This  table presents the  time-average (across $480-2T$ windows) of the time series for the annualized Sharpe Ratios  
         both in-sample (bottom panel) and out-of-sample (top panel) for the Tangency Portfolio,
    correspondingly to  the (time-varying) estimated number of factors $\tilde{r}$   (column two), described in Section~\ref{emp_number_tv}, and to the values  one to $ 10 $ (columns 3-12), respectively.  The local-PCA is evaluated with $T=24$.  The local-PCA is evaluated with $T=24$. The constant-PCA results do not depend on $T=120$, and are identical to the $T=24$ table. Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table}
\label{Table5OAT120}
\end{table}

\bibliography{Biblio_PCA}


\end{document}
